Fuzzy Approach to the Capacitated Facility Location Problem

Lýdia Gábrišová

doi:10.2478/v10198-011-0009-8

Abstract

The main objective of the paper is to devise and implement the algorithmic solution of problem of optimal location of set of facilities so that all demands of customers will be satisfied and the relevant costs of creating and maintenance of facilities are minimal. This optimization task can be modelled by capacitated facility location problem, which is the special class of integer linear programming problems. The output parameters of solution of such problems are the number of facilities actually used, the location of facilities and the assignment of individual customers to the concrete facility from the optimal set of facilities. When a distribution system is to be designed, limits on terminal capability often must be taken into account. The capacity constraints in this case and also in other problems dealing with facility locations cause the severe difficulties in exact solving procedures because the underlying mathematical models are NP-hard. One possible approach is approximate method based on Lagrangean relaxation of the capacity constraints, which has several advantageous properties. In capacitated location problems, the capacity of a facility as an upper limit of its ability to satisfy a given volume of demands cannot be precisely determined in most of practical applications. This circumstance evokes an idea to employ fuzzy approach for handling of the capacities and to utilize suggests an idea the fuzzy description in capacity constraint relaxation. The relaxed problem is exactly solvable even for real-world instances. It has been used in the heuristic method exploring the concept of measure of infeasibility.

Highlights

In contrast to uncapacitated facility location problem, which can be solved exactly for real-sized instances containing hundreds of possible locations and thousands of customers, the capacitated location problem resists to all attempts to solve it exactly in a reasonable time
The suggested technique for solving capacitated location problem with uncertain capacity limitations using the theory of fuzzy sets gives results shown in Table 3 for all 9 chosen problems
If there were an exact method for solving the crisp problem (1) – (5), several changes in graph of illustrated curves would not occur

Summary

SOLVING METHOD

Cost optimal design of the most of distribution and servicing systems consists in decisions on number and locations of facilities, from which demands of customers are satisfied. In contrast to uncapacitated facility location problem, which can be solved exactly for real-sized instances containing hundreds of possible locations and thousands of customers, the capacitated location problem resists to all attempts to solve it exactly in a reasonable time To avoid this complication and to obtain good solution of the capacitated facility location problem in a sensible time, we employ a fuzzy approach to handle the capacity limits and utilize an algorithm based on Erlenkotter’s approach [1]. This approach relaxes the capacity constraints making use of a penalty function derived from membership functions describing customer clusters for individual facility locations. When solving this task we will use fuzzy logic with the theorem that the task can be solved for deterministically stated data

Mathematical model of problem

Approach with exceeding of given capacities

NUMERICAL EXPERIMENTS

CONCLUSIONS